Algebra 2
Notes
1.3 Transformations of Function Graphs
Date: _______________
We can change the position, size, and orientation of graphs using different transformations. How a
graph is transformed is determined by the way certain numbers, called parameters, are introduced in
the function.
Investigating Translations
Complete the following given the graph of f(x):
Let 𝑔(𝑥) = 𝑓(𝑥) + 𝑘, where k is the parameter.
A. Let k = 4, so 𝑔(𝑥) = 𝑓(𝑥) + ______. Fill in the values in the
table below to graph the function g(x). What translation
occurs?
x
-1
1
3
5
f(x)
f(x) + 4
What translation do you think would occur if k was negative, so 𝑔(𝑥) = 𝑓(𝑥) − 𝑘?
Let 𝑔(𝑥) = 𝑓(𝑥 − ℎ), where h is the parameter.
B. Let h = 2, so 𝑔(𝑥) = 𝑓(𝑥 − ____). Fill in the mapping
diagram by determining what inputs for g produce the
inputs for f after you subtract 2. What translation occurs
when h is positive?
What translation do you think would occur if h was negative, so 𝑔(𝑥) = 𝑓(𝑥 − (−ℎ)) = 𝑓(𝑥 + ℎ)?
Reflect. You can transform the graph of f(x) to obtain the graph 𝑔(𝑥) = 𝑓(𝑥 − ℎ) + 𝑘 by combining
transformations. Predict what will happen by completing the table below.
Sign of h
+
+
-
Sign of k
+
+
-
Transformations of the Graph f(x)
Translate right h units and up k units
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Algebra 2
Notes
Investigating Stretches and Compressions
Let 𝑔(𝑥) = 𝑎 ∙ 𝑓(𝑥), where a is the parameter. Determine the effect the value of a has on each
graph.
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A. Let a= 2, so 𝑔(𝑥) = ____𝑓(𝑥).
x
-1
1
3
5
f(x)
-2
2
-2
2
B. Let 𝑎 = 2, so 𝑔(𝑥) = ____𝑓(𝑥)
2f(x)
x
f(x)
-1
1
3
5
-2
2
-2
2
𝟏
f(x)
𝟐
1
Let 𝑔(𝑥) = 𝑓(𝑏 ∙ 𝑥), where p is the parameter. Determine the effect the value of b has on each
graph.
1
C. Let b = 2, so 𝑔(𝑥) = 𝑓(2 ∙ 𝑥)
1
D. Make a conjecture: How would you expect the graph of 𝑔(𝑥) = 𝑓( ∙ 𝑥) to be related to the graph of
f(x) when b is a number between 0 and 1?
𝑏
Stretches and Compressions
Vertical
𝑔(𝑥) = 𝑎 ∙ 𝑓(𝑥)
Horizontal
𝑔(𝑥) = 𝑓
1
𝑏
∙𝑥
2
Algebra 2
Notes
Investigating Reflections
A. Let a = -1, so 𝑔(𝑥) = −𝑓(𝑥)
x
-1
1
3
5
f(x)
-2
2
-2
2
B. Let b = -1, so 𝑔(𝑥) = 𝑓(−𝑥)
x
-1
1
3
5
-f(x)
Reflection across the __________________
f(x)
-2
2
-2
2
-x
Reflection across the __________________
Even and Odd Functions
Reflections also tell us whether a function is even or odd.
Even Function
Odd Function
-
A function for which 𝑓(−𝑥) = 𝑓(𝑥)
-
A function for which −𝑓(𝑥) = 𝑓(−𝑥)
-
Symmetric across the _______________
-
Reflections across both axes yield the
same graph
Putting it All Together
𝑔(𝑥) = 𝑎 ∙ 𝑓(𝑥 − ℎ) + 𝑘
𝑔(𝑥) = 𝑓
1
(𝑥
𝑏
− ℎ) + 𝑘
3
Algebra 2
Notes
Transforming the Graph of the Parent Quadratic Function
Learning Target G: I can transform a graph of the parent quadratic function.
Parent Quadratic Function: 𝑓(𝑥) = 𝑥 2 .
The graph of a quadratic function is called a ____________________
with vertex at _____________ and axis of symmetry at _____________.
Reference Points: (-1, 1) (0, 0) (1, 1)
Describe how to transform the graph of 𝒇(𝒙) = 𝒙𝟐 to obtain the graph of the related function
g(x). Then, draw the graph of g(x).
A. 𝑔(𝑥) = −3𝑓(𝑥 − 2) − 4
B. 𝑔(𝑥) = 𝑓
1
2
(𝑥 + 5) + 2
Reflect. Is the function 𝑓(𝑥) = 𝑥 2 an even function, odd function, or neither? Explain.
4
Algebra 2
Notes
Your Turn. Describe how to transform the graph 𝑓(𝑥) = 𝑥 2 to obtain the graph of the related
function 𝑔(𝑥) = 𝑓(−4(𝑥 − 3)) + 1. Then, draw the graph of g(x).
Modeling with Quadratic Functions
Learning Target H: I can model with a quadratic function.
We can use quadratic functions to model various real- life situations. In order to do this, we must
identify the parameters. Horizontal stretches and compression can be expressed as vertical, too, so we
only need to find the parameters in the function 𝑔(𝑥) = 𝑎 ∙ 𝑓(𝑥 − ℎ) + 𝑘.
Example)
5
Algebra 2
Notes
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